Finite Group Databasebeta

← #30620783

Subgroups — #30620783

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462512C211yesmaximal, characteristic
2#21462510C211yesmaximal, characteristic
3#21461928C211yesmaximal, characteristic, complemented
4#21461927C211yesmaximal, characteristic, complemented
5#21461887C211yesmaximal, characteristic, complemented
6#21461886C211yesmaximal, characteristic, complemented
7#21461885C211yesmaximal, characteristic, complemented
8#21461884C211yesmaximal, characteristic, complemented
9#21461727C211yesmaximal, characteristic
10#21461725C211yesmaximal, characteristic
11#21461700C211yesmaximal, characteristic
12#21461627C211yesmaximal, characteristic, complemented
13#21461622C211yesmaximal, characteristic, complemented
14#21461559C211yesmaximal, characteristic
15#21454406C211yesmaximal, characteristic
16#16840787—21no—
17#16840787—21no—
18#16840786—21no—
19#16840786—21no—
20#16840784—21no—
21#16840783—21no—
22#16840780—21no—
23#16840779—21no—
24C2×#14519848C2211yescharacteristic
25C2×#14519847C2211yescharacteristic
26#16840769C2211yescharacteristic
27#16840769C2211yescharacteristic
28#16840769C2211yescharacteristic
29#16840769C2211yescharacteristic
30#16840733C2211yescharacteristic
31#16840733C2211yescharacteristic
32D61⋊(C22⋊C8)C2211yescharacteristic
33#16840727C2211yescharacteristic
34#16840727C2211yescharacteristic
35#16840727C2211yescharacteristic
36#16840727C2211yescharacteristic
37C2×#14519831C2211yescharacteristic
38C2×C8×#6923C2211yescharacteristic
39C4×C22⋊#6923C2211yescharacteristic
40#16840638C2211yescharacteristic
41#16840638C2211yescharacteristic
42#16840632C2211yescharacteristic
43#16840632C2211yescharacteristic
44#16840632C2211yescharacteristic
45#16840632C2211yescharacteristic
46#16840632C2211yescharacteristic
47#16840632C2211yescharacteristic
48#16840632C2211yescharacteristic
49#16840632C2211yescharacteristic
50(C2×#6923)⋊C8C2211yescharacteristic
51(C2×#6923)⋊C8C2211yescharacteristic
52C2×M16×D61C2211yescharacteristic
53C2×#14519735C2211yescharacteristic
54#117×D61C2211yescharacteristic
55#16839805C411yescharacteristic, complemented
56#16839804C411yescharacteristic, complemented
57#16839803C411yescharacteristic, complemented
58#16839803C411yescharacteristic, complemented
59#16839802C411yescharacteristic, complemented
60#16839801C411yescharacteristic, complemented
61#16839801C411yescharacteristic, complemented
62#16839799C2211yescharacteristic
63#16839799C2211yescharacteristic
64#16839797C2211yescharacteristic
65#16839797C2211yescharacteristic
66C61×#428C411yescharacteristic, complemented
67C2×#11758213C2311yescharacteristic
68C2×#11758213C2311yescharacteristic
69C2×#11758213C2311yescharacteristic
70C2×#11758213C2311yescharacteristic
71C2×D61⋊C8C2311yescharacteristic
72C2×D61⋊C8C2311yescharacteristic
73C2×C22⋊#6923—21no—
74C2×C4×#6923C2311yescharacteristic
75C2×C4×#6923C2311yescharacteristic
76#14519848—22no—
77#14519847—22no—
78#14519844—21no—
79#14519844—21no—
80#14519844—21no—
81#14519844—21no—
82C4⋊#92420—21no—
83C4⋊#92420—21no—
84C4⋊#92420—21no—
85C4⋊#92420—21no—
86#14519834—21no—
87#14519834—21no—
88#14519834D411yescharacteristic
89#14519834D411yescharacteristic
90C22⋊#92420—21no—
91C22⋊#92420—21no—
92C22⋊#92420D411yescharacteristic
93C22⋊#92420D411yescharacteristic
94#14519832—21no—
95#14519832—21no—
96#14519832—21no—
97#14519832—21no—
98#14519831—22no—
99C4×#92420—21no—
100C4×#92420—21no—
101C4×#92420—21no—
102C4×#92420—21no—
103C8×#6923—22no—
104(C2×#6923)⋊C4—21no—
105C22×C4×D61C2311yescharacteristic
106C2×#11758194C2×C411yescharacteristic
107C2×#11758189C2×C411yescharacteristic
108C2×#11758189C2×C411yescharacteristic
109M16×D61—22no—
110M16×D61—22no—
111C2×C8×D61C2311yescharacteristic
112C2×C8×D61C2311yescharacteristic
113#14519736C2×C411yescharacteristic
114#14519735—21no—
115#14519735—21no—
116#14519735C2×C411yescharacteristic
117#14519735C2×C411yescharacteristic
118#14519735C2311yescharacteristic
119#14519735C2311yescharacteristic
120#14519735C2311yescharacteristic
121#14519735C2311yescharacteristic
122#14519734—21no—
123#14519733—21no—
124C2×C4⋊Dic61C2×C411yescharacteristic
125C4⋊C4×D61C2×C411yescharacteristic
126C4⋊C4×D61C2×C411yescharacteristic
127C2×#11758198—21no—
128C22⋊C4×D61—21no—
129C2×C4×Dic61—21no—
130C42×D61—21no—
131#14519711C2×C411yescharacteristic
132#14519711C2×C411yescharacteristic
133#14519710C2×C411yescharacteristic
134#14519710C2×C411yescharacteristic
135#14519710C2×C411yescharacteristic
136#14519710C2×C411yescharacteristic
137#14519709C2×C411yescharacteristic
138#14519709C2×C411yescharacteristic
139#14519708C2×C411yescharacteristic
140#14519708C2×C411yescharacteristic
141#14519708C2×C411yescharacteristic
142#14519708C2×C411yescharacteristic
143C2×M16×C61C2×C411yescharacteristic
144#117×C61C2×C411yescharacteristic
145C2×C4⋊C4×C61C2×C411yescharacteristic
146#107×C61C2×C411yescharacteristic
147#107×C61C2×C411yescharacteristic
148#106×C61C2×C411yescharacteristic
149#106×C61C2×C411yescharacteristic
150C23×D61#4111yescharacteristic
151C22×#6923—21no—
152C22×#6923—21no—
153C22×C4×C61—21no—
154C22×C4×C61C22×C411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61C2×D411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61C22×C411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61#4111yescharacteristic
166C2×C4×D61C2411yescharacteristic
167C22⋊#6923—42no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420C2×D411yescharacteristic
173C2×#92420C2×D411yescharacteristic
174C2×#92420C2×D411yescharacteristic
175C2×#92420C2×D411yescharacteristic
176C4×#6923—21no—
177C4×#6923—21no—
178C4×#6923—21no—
179C4×#6923—21no—
180#11758213—21no—
181#11758213—21no—
182#11758213—21no—
183#11758213—21no—
184#11758213—21no—
185#11758213—21no—
186#11758213—21no—
187#11758213—21no—
188D61⋊C8—21no—
189D61⋊C8—21no—
190D61⋊C8—21no—
191D61⋊C8—21no—
192M16×C61—22no—
193C2×C8×C61C22×C411yescharacteristic
194C2×C8×C61C22×C411yescharacteristic
195C4⋊C4×C61—21no—
196C4⋊C4×C61C22×C411yescharacteristic
197C4⋊C4×C61C22×C411yescharacteristic
198C4⋊C4×C61C22×C411yescharacteristic
199C4⋊C4×C61C22×C411yescharacteristic
200C22⋊C4×C61—21no—
201C42×C61—21no—
202C22⋊Dic61—21no—
203#11758198—21no—
204#11758198—21no—
205#11758198—21no—
206#11758198—21no—
207#11758198—21no—
208#11758198—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61C22×C411yescharacteristic
211C4⋊Dic61C22×C411yescharacteristic
212C4⋊Dic61C22×C411yescharacteristic
213C4⋊Dic61C22×C411yescharacteristic
214#11758196—21no—
215#11758196—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223#11758194—22no—
224C2×#92418C22×C411yescharacteristic
225C2×#92418C22×C411yescharacteristic
226#11758189—22no—
227#11758189—22no—
228C8×D61—21no—
229C8×D61—21no—
230C8×D61#4112yes—
231C8×D61#4112yes—
232C23×C61C4×D411yescharacteristic
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61C4×D411yescharacteristic
238C22×D61C2×#4111yescharacteristic
239C2×#6923—22no—
240C2×#6923—41no—
241C2×#6923—41no—
242C2×#6923—41no—
243C2×#6923—41no—
244C2×#6923—41no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—81no—
252C2×C4×C61C4×D411yescharacteristic
253C2×C4×C61C23×C411yescharacteristic
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—42no—
261C2×Dic61C4×D411yescharacteristic
262C2×Dic61C22×D411yescharacteristic
263C4×D61—21no—
264C4×D61—21no—
265C4×D61—21no—
266C4×D61—21no—
267C4×D61—21no—
268C4×D61—21no—
269C4×D61—42no—
270C4×D61—81no—
271C4×D61C2×#4111yescharacteristic
272C4×D61C2×#4111yescharacteristic
273C4×D612−1+411yescharacteristic
274C4×D612−1+411yescharacteristic
275#92420—21no—
276#92420—21no—
277#92420—21no—
278#92420—21no—
279#92420—41no—
280#92420—41no—
281#92420—41no—
282#92420—41no—
283C8×C61—21no—
284C8×C61#11712yes—
285#92418—21no—
286#92418#11712yes—
287#22608—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61C2×C4×D411yescharacteristic
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61#55211yescharacteristic
300C2×D61#55211yescharacteristic
301#6923—42no—
302#6923—81no—
303C4×C61—21no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61C2×#11711yescharacteristic
309C4×C61#51911yescharacteristic
310Dic61—21no—
311Dic61—41no—
312Dic61—41no—
313Dic61—41no—
314Dic61—41no—
315Dic61#54711yescharacteristic
316Dic61#54711yescharacteristic
317#2924—611no—
318#2924—611no—
319#2911—611no—
320#1946—611no—
321#1945—611no—
322#1922—611no—
323#1922—611no—
324#1922—611no—
325#1922—611no—
326#1826—611no—
327#1826—611no—
328#1811—611no—
329#1773—611no—
330#1772—611no—
331#1753—611no—
332C2×C61—41no—
333C2×C61#234411yescharacteristic
334C2×C61#295511yescharacteristic
335C2×C61#295611yescharacteristic
336D61—21no—
337D61—21no—
338D61—41no—
339C22×M16—611no—
340C2×#117—611no—
341C2×#117—611no—
342#433—1221no—
343#433—1221no—
344#433—1221no—
345#433—1221no—
346#432—1221no—
347#432—1221no—
348#431—1221no—
349#431—1221no—
350C2×#108—611no—
351C2×#108—611no—
352#428—611no—
353#428—611no—
354#428—611no—
355#428—611no—
356#428—611no—
357#428—611no—
358#428—611no—
359#428#692311yescharacteristic, complemented
360#427—611no—
361#427—611no—
362#427—611no—
363#427—611no—
364#424—611no—
365#424—611no—
366#424—611no—
367#424—611no—
368#407—611no—
369#407—611no—
370C2×C22⋊C8—611no—
371#405—611no—
372#405—611no—
373#405—611no—
374#405—611no—
375C2×#97—611no—
376C2×C4×C8—611no—
377C4×C22⋊C4—611no—
378#341—611no—
379#341—611no—
380#339—611no—
381#339—611no—
382#339—611no—
383#339—611no—
384#339—611no—
385#339—611no—
386#339—611no—
387#339—611no—
388#336—611no—
389#336—611no—
390C61#2260811yescharacteristic, complemented, Sylow
391C23×C4—611no—
392C2×M16—611no—
393C2×M16—611no—
394C2×M16—611no—
395C2×M16—611no—
396C2×M16—611no—
397C2×M16—611no—
398C2×M16—611no—
399C2×M16—1222no—
400C2×M16—1222no—
401C2×M16C2×#692311yescharacteristic
402C22×C8—611no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8—611no—
406#117—611no—
407#117—611no—
408#117—611no—
409#117—611no—
410#117—611no—
411#117—611no—
412#117—611no—
413#117—1221no—
414#117—1221no—
415#117—1221no—
416#117—1221no—
417#117C2×#692311yescharacteristic
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4C2×#692311yescharacteristic
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42—1221no—
429#108—1222no—
430#108—1222no—
431#107—611no—
432#107—611no—
433#107—611no—
434#107—611no—
435#107—611no—
436#107—611no—
437#107C2×#692311yescharacteristic
438#107C2×#692311yescharacteristic
439#106—611no—
440#106—611no—
441#106—611no—
442#106—611no—
443#106—611no—
444#106—611no—
445#106C2×#692311yescharacteristic
446#106C2×#692311yescharacteristic
447C4⋊C8—1221no—
448C4⋊C8—1221no—
449C4⋊C8—1221no—
450C4⋊C8—1221no—
451C4⋊C8—1221no—
452C4⋊C8—1221no—
453C4⋊C8—1221no—
454C4⋊C8—1221no—
455C22⋊C8—611no—
456C22⋊C8—611no—
457C22⋊C8—611no—
458C22⋊C8—611no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461C22⋊C8—1221no—
462C22⋊C8—1221no—
463#97—1222no—
464#97—1221no—
465#97—1221no—
466#97—1221no—
467#97—1221no—
468C4×C8—1222no—
469C4×C8—1221no—
470C4×C8—1221no—
471C4×C8—1221no—
472C4×C8—1221no—
473#95—1221no—
474C24—611no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4C22×#692311yescharacteristic
491M16—22no—
492M16—1222no—
493M16—1222no—
494M16—1222no—
495M16—1221no—
496M16—1221no—
497M16—1221no—
498M16—1221no—
499M16—1221no—
500M16—1221no—
501M16—1221no—
502M16—1221no—
503C2×C8—611no—
504C2×C8—612no—
505C2×C8—612no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—611no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8C22×#692311yescharacteristic
522C2×C8C22×#692311yescharacteristic
523C4⋊C4—21no—
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—611no—
527C4⋊C4—611no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4C22×#692311yescharacteristic
532C4⋊C4C22×#692311yescharacteristic
533C4⋊C4C22×#692311yescharacteristic
534C4⋊C4C22×#692311yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—1221no—
542C22⋊C4—1221no—
543C22⋊C4—2442no—
544C42—21no—
545C42—1221no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42—1221no—
552C42—1221no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C23—611no—
557C23—611no—
558C23—1221no—
559C23—1221no—
560C23—2441no—
561C23—2441no—
562C23D4×#692311yescharacteristic
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—81no—
570C2×C4—611no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1222no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—4881no—
590C2×C4—2442no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2442no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4D4×#692311yescharacteristic
598C2×C4C23×#692311yescharacteristic
599C8—21no—
600C8—612no—
601C8—1221no—
602C8—1221no—
603C8—1221no—
604C8—1221no—
605C8—1221no—
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C8#1451985612yes—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—611no—
615C22—611no—
616C22—1221no—
617C22—1221no—
618C22—1221no—
619C22—1221no—
620C22—1221no—
621C22—4881no—
622C22—4881no—
623C22C2×D4×#692311yescharacteristic
624C4—21no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—611no—
630C4—611no—
631C4—1221no—
632C4—4881no—
633C4—2442no—
634C4—2441no—
635C4—2441no—
636C4—2441no—
637C4—2441no—
638C4C2×#1451985611yescharacteristic
639C4#1684083011yescharacteristic
640C2—41no—
641C2—1221no—
642C2—1221no—
643C2—2441no—
644C2#2146232711yescharacteristic
645C2#2146259211yescharacteristic
646C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#428C4
socleC22×C61C2×C4×D4
prime radicalC23×D61#41
cosocleC2×C4×D61C24
hypercenter#428#6923
hypocenterC61#22608
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #21461928C2semidirect
4. #21461927C2semidirect
5. #21461887C2semidirect
6. #21461886C2semidirect
7. #21461885C2semidirect
8. #21461884C2semidirect
12. #21461627C2semidirect
13. #21461622C2semidirect
55. #16839805C4semidirect
56. #16839804C4semidirect
57. #16839803C4semidirect
58. #16839803C4semidirect
59. #16839802C4semidirect
60. #16839801C4semidirect
61. #16839801C4semidirect
66. C61×#428C4semidirect
359. #428#6923semidirect
390. C61#22608semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214625122
#214625102
#214619282
#214619272
#214618872
#214618862
#214618852
#214618842
#214617272
#214617252
#214617002
#214616272
#214616222
#214615592
#214544062
#2260861